Some of the first evidence of the use of two-dimensional time frequency representations of communication signals was presented in a paper by D. Gabor entitled "Theory of Communication", Journal IEE (London) 93 (III), pp. 429-457, (1946). Shortly thereafter, J. Ville wrote a paper entitled "Theories et Application de la notion de signal analytique", Cables et Transmissions 2A (1) pp. 61-74, (1948), in which his approach is based on the space/momentum representation of quantum mechanical variables introduced in a paper by E. P. Wigner entitled "On the Quantum Correction for Thermo-Dynamic Equilibrium", Phys. Review 40, pp. 749-759, (1932). Generally speaking, the time/frequency distribution was introduced because of the inability of classical representations such as correlation functions or spectral density functions to display information contained in non-stationary signals. These non-stationary signals whose spectral content varies with time are of major engineering importance and the analyses of these signals to enable easy interpretation has obvious advantageous.
Totally different signals may have the same spectral density. This anomaly is attributed to the fact that the information which differentiates these signals is contained in the phase spectrum, as will be elaborated on below. The localization in time of a particular frequency of the signal is measured by the group delay function .tau..sub.g (f), which is the derivative of the phase spectrum, ##EQU3## where .theta.(f) is the phase spectrum of the signal.
The phase spectrum is an integral part of the representation of the signal. Removing this information will lead to an effective scrambling of the signal's frequency information. Therefore, the spectral density which represents the distribution of signal energy over the frequency axis must, in general, be accompanied by the time delay .tau..sub.g (f) which localizes in time the particular frequency of the signal, if the representation of the signal is to make sense.
In like manner the instantaneous power of a signal [s(t)].sup.2 which represents the distribution of signal energy over the time axis alone cannot characterize the signal, but must be coupled with the instantaneous frequency f.sub.i (t) of the signal which localizes in time the particular frequency of the signal.
This instantaneous frequency f.sub.i (t) was defined in the above identified article by Ville as: ##EQU4## where z(t)=a(t)e.sup.j .phi..sup.(t) is the analytic signal associated with the real signal s(t) as follows: ##EQU5## where H is the Hilbert transform H[s(t)] ##EQU6##
As suggested, in the Gabor article, the signals can be represented in a wider space that would combine all this information in a more accessable manner. Such a representation would for example, provide a distribution of signal energy E(t,f) versus both time and frequency. A first approach adopted by many authors was to set out a "discretization" of the time domain. A window p(t-t.sub.o) is applied to the real signal s(t) and the magnitude squared of the Fourier transform is calculated. By varying t in multiples of an increment, .DELTA., a spectral density is obtained that is a function of f and t.sub.o. Information that was otherwise lost when the spectral density of the whole signal is used may now be retrieved by noting the position t.sub.o of the window (the increment .DELTA.) with a resolution as determined by the width of this window. For an optimum analysis based on this method the window width .DELTA. is selected which satisfies the definition: ##EQU7##
As may be apparent from this analysis a basic problem arises in that this method requires an a-priori knowledge of the instantaneous frequency f.sub.i (t), which can only be obtained after a time/frequency analysis. It is obvious that a successful procedure based on this method must be iteractive.
Another equivalent approach consists of the "discretizing" of the frequency domain, by passing the signal through a bank of adjacent filters centered around frequencies nf and with a bandwidth .DELTA.(f), and applying an implicit windowing P(f-f.sub.o) to the Fourier transform S(f) of the signal. Bouachache et al showed in their article entitled "A Necessary and Sufficient Condition for the Positivity of Time Frequency Distributions", Comptes Rendus Acad. des Sciences, Paris, 288, Series A, pp. 307-309, (1979) that these time/frequency representations are equivalent if P(f) is the Fourier transform of p(t).
A general class of time/frequency distributions was proposed in an article by Cohen entitled "Generalized Phase Space Distributions", Journal of Mathematical Physics, 7, pp. 781-786, (1966). In a later study Bouachache, Representation tempsfrequence, PhD. Thesis University of Grenoble 1982 showed that among Cohen's general class, the Wigner-Ville Distribution (WVD) performs best when applied to the analysis of modulated signals. A thorough analysis of the Bouachache study is set forth in the article entitled "Sur la Possibilite d'utiliser la Representation Conjointe en temps et Frequence dans l'analyse des Signaux Modules en Frequence emis en Vibrosismique", Colloque Nat. Trait, Sign., 7th GRETSI, pp. 121:1-121:6, (1979).
The WVD is expressed as equation 6 ##EQU8##
A major significance of Cohen's formula for signal analysis is that any useful time/frequency distribution is the result of a two-dimensional smoothing of the WVD, so that a measure of the localization in time and frequency of the time/frequency distribution is relative to the WVD. The above cited articles of Bouachache also established that the WVD of a signal is concentrated along its instantaneous frequency f.sub.i (t) with a resolution that is a function of the BT product of the signal.